The present invention generally relates to techniques for controlling an electromechanical actuator and, more particularly, is concerned with an H.sub..infin. controller for improving the operating characteristics of an electro-mechanical actuator. Specifically, the uses a robust control design technique called H.sub..infin. synthesis to design a controller to shape the frequency response characteristics to improve stability and performance characteristics of an electro-mechanical actuator. The following discussion of H.sub..infin. control design theory as applied to and electro-mechanical actuator is from a published paper AIAA-92-4314-CP "H.sub..infin. Controller Design for an SISO Electromechanical Actuator" presented to the American Institute of Aeronautics and Astronautics (AIAA) Guidance, Navigation and Control Conference on Aug. 9, 1992
H.sub..infin. robust control design allows shaping of the frequency response of the feedback characteristics of a system by increasing the system gain in the lower frequency bands for performance enhancement while suppressing gains at the higher frequencies so as not to excite structural dynamics. Heretofore, H.sub..infin. design has been applied to a multiple-input multiple-output control system, such as in the case of autopilot design, to achieve a high performance but robustly stable design. The H.sub..infin. implementation has required sophisticated microprocessor chips with the memory capacity to handle all of the states in the state space model.
In the past, the bandwidth of actuators were limited to that which would not excite structural modes and the existing autopilot designs could not utilize larger a bandwidth anyway. Improvements in autopilot designs using modern control design theory allow the utilization of better performing actuators. When structural modes are found to be excited by the actuators usually a filter is placed either in the feedback or feedforward path which introduces a phase lag to reduce the responsiveness of the control system to an actuator with too large a bandwidth.
As briefly noted above, the H.sub..infin. control theory involves singular value loop shaping of the sensitivity and complementary sensitivity matrices based on the singular value Bode plot. Ordinarily, H.sub..infin. control theory is used for multivariable-input multivariable-output (MIMO) systems but can be applied to a single-input single-output (SISO) disclosure. The singular values of a matrix is defined as the positive square roots of the eigenvalues of the product of the matrix and its transpose. The transfer function sensitivity matrix S and is given by EQU S(s)=[I(n)+L(s)].sup.-1
were I(n) is the identity matrix or unity matrix of size (n.times.n) with 1's on the main diagonal and 0's elsewhere, "s" is the Laplacian operator equal to j*omega, j is .sqroot.(-1), omega is the frequency in radian/sec and L(s) is the loop gain matrix. A large loop L(s) means that the sensitivity matrix will have small singular values so that a disturbance at the actuator output A will have little effect on the error input to the H.sub..infin. controller. Good disturbance rejection is highly desirable for a controller. The complementary sensitivity matrix is the transfer function from I to actuator output A or the system transfer function given by EQU T(s)=L(s)*[I(n)+L(s)].sup.-1 =I(n)-S(s)
A small value of loop gain L(s) means that T(s) is small while a large loop gain L(s) will drive T(s) to the identity matrix while S(s) goes small.
The term H.sub..infin. derives from taking the infinity norm of the maximum singular values of a matrix over all frequencies. If a matrix has size n, there are n eigenvalues. The singular values of a matrix G are defined as the non-negative square roots of the eigenvalues of the product of matrix G and its transpose G*. The most useful property of the maximum singular value concept is that the maximal "gain" of a matrix over all frequencies is given by the peak of the maximum singular value over all frequencies. The H.sub..infin. norm, .parallel.G.parallel..sub..infin., is thus defined by EQU .parallel.G.parallel..sub..infin. =sup.sub..omega. (o.sub.max (G(j.omega.))
where sup is the least upper bound over all frequencies .omega.. Assuring that the infinity norm of G is &lt;=1 over all frequencies means it is robust and will not be unstable under any foreseeable conditions.
The H.sub..infin. control design is performed in the frequency domain with specifications on performance governing the frequency loop shaping of the sensitivity matrix and robustness governing the frequency loop shaping of the complementary sensitivity matrix. If one was not worried about exciting structural modes or unmodeled high frequency dynamics, it would be logical to have as high a loop gain as possible over the whole frequency range. The robustness specification .vertline.W.sub.3 (s).sup.-1 .vertline. suppresses the maximum singular values at the higher frequencies for the complementary sensitivity matrix or system transfer function so as not to excite unmodeled plant dynamics or fundamental structural frequencies. The performance specification .vertline.W.sub.1 (s).sup.-1 .vertline. suppresses the sensitivity matrix maximum singular values at the lower frequencies for good disturbance rejection and minimization of steady-state error through high low frequency loop gains. The 0 Db or magnitude=1 crossover frequency of the performance specification must be sufficiently below the 0 Db frequency crossover of the robustness specification for a H.sub..infin. solution to exist. Commercial software in the Mathwork's MATLAB Robust Control Toolbox for H.sub..infin. design were utilized in the design example in the AIAA paper cited. An iterative procedure was followed in which the loop gain was increased for a given performance specification which drove the maximum singular value of the sensitivity matrix to the performance boundary and simultaneously drove the maximum singular value of the complementary sensitivity matrix to the robustness boundary at a higher frequency range.